Laws of Algebra of Sets

Have you got a favourite dish? Perhaps it’s a cheeseburger from your favourite fast-food joint. A cheeseburger, French fries, a drink, ketchup packets, and napkins are most often included in this meal. This is referred to as a set-in in real life.

According to the technical definition, a set is a collection of precisely particular objects. Let’s go over the general properties and principles of Laws of Algebra of Sets.

Set Theory

A set is a collection of well-defined elements that do not differ from one person to another. It can be represented in two ways: as a set builder or a roster. Curly braces {} are commonly used to represent sets. The empty set, finite set, singleton set, infinite set, equivalent set, disjoint sets, equal sets, subsets, supersets, and universal sets are examples of distinct types of sets.

Any collection of items must meet specified criteria and be carefully defined. For example, one person may believe a cheeseburger dinner from a fast-food restaurant is fantastic, while another may find it repulsive, rendering the criteria ineffective. Edible dishes that include bread are an example of a valid set to include the cheeseburger dinner.

Understanding of Laws of Algebra of Sets

Here’s an example of the laws of the algebra of sets.

Let’s build a list of foods and see which ones qualify for a group of edibles that includes bread, which we’ll call “S.”

S is equal to a sandwich, hamburger, burger (cheese), toast, bread}

The symbol ∈ denotes that something belongs to a group. Grilled cheese ∈ S, for example, refers to the fact that it comes under the set S. Because grilled cheese is a sandwich, this is correct. Ice cream ∉ S denotes that ice cream is not included in set S because it lacks bread.

Let’s have a look at some set properties. It makes no difference what order the items in a set are in. We may put toast first and sandwiches last in our list of edible foods that incorporate bread. Count once if an item in a set is repeated. Let’s imagine we have a set of W letters representing the letters in the word.

”cheeseburger”.

So, the outcome would be: W is equal to {c, h, e, e, s, e, b, u, r, g, e, r}. As there are 4 e’s and 2 r’s, you can even write it as W = {c, h, e, s, b, u, r, g}.

The Laws of Sets

Let’s look at the different Laws of Algebra of Sets at a time.

1. Union of Sets

Assume we have two sets of items:

S  = { sandwich, burger (ham), burger (cheese), toast, bread pudding}, and

 B = {hamburger cheeseburger}. 

We’ll refer back to these sets throughout the rest of the lesson. All objects are part of both sets or make up the union of two sets. Sets S and B are combined as 

S ∪ B = {sandwich, hamburger, cheeseburger, toast, bread}, which comprises all the matters in mutual sets.

2. Intersection of Sets

What is common to both sets is defined by the intersection of sets. The hamburger and cheeseburger, for example, are shared by both groups S and B.

 S ∩ B = {hamburger, cheeseburger} is the intersection of these sets. A Venn diagram of the two sets is similar to this notation.

Both sets include hamburgers and cheeseburgers, hence, they intersect the two sets.

3. Commutative Law

Because 4 + 3 = 7 and 3 + 4 = 7, addition is a commutative property, the order in which the numbers are added makes no difference. This set is no exception.

4. Set Difference

A difference between sets is a set operation that involves subtracting members from a set, which is related to the difference between numbers. All the elements that are in set A but not in set B are listed in the difference between sets A and B, marked as A B. Consider the following example to understand the set operation of set difference better:

 If A = {1, 2, 3, 4 }and B ={ 3, 4, 5, 7}, then A – B ={ 1, 2} is the difference between sets A and B.

5. Complement of Sets

The complement of a set A indicated as A′ or Ac (read as A complement) is the set of all elements in the specified universal set(U) that are not present in set A. Consider the following example to better understand the complement of sets set operation:

 If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, then A’ ={ 5, 6, 7, 8, 9} is the complement of set A.

Conclusion

The concept of set operations is akin to fundamental operations on numbers. In mathematics, a set is a group of objects, such as numbers, alphabets, or any other real-world objects. There are occasions when building a relationship between two or more sets is necessary. Then there’s the idea of set operations. You can refer to the laws of